Advertisement

Archive of Applied Mechanics

, Volume 86, Issue 4, pp 669–686 | Cite as

A damage tolerance analysis for complex structures

  • Daniel Pucknat
  • Robert LiebichEmail author
Original
  • 212 Downloads

Abstract

The assessment of damage is usually performed to determine safety or residual life, which is necessary to define inspection intervals for instance. The concept of a damage tolerance analysis presented in this paper ties on and tries to reveal the possible redundancy that statically indeterminate complex structures provide. Classic strength assessments are not able to map changes in load paths or structural supporting effects and that might lead to a huge waste of potential in weight reduction. The damage tolerance analysis is applied to an exemplary complex structure and shows that the fatigue driven damages are not critical for further operation at the assessed load level. The concept uses a new method for the calculation of equivalent stress—the ‘Modified Mohr Mises’ (MMM)-Hypothesis. The MMM-Hypothesis allows the assessment of non-proportional stresses within a fully automated process, due to its invariant equivalent stress notation. Those non-proportional stresses are most common in complex structures of aircrafts, spacecrafts and vehicles for instance.

Keywords

Damage tolerance Fatigue Crack propagation MMM-Hypothesis Mises-Hypothesis Non-proportional loading 

List of symbols

\(\varepsilon \)

Strain

\(\sigma _{\mathrm{eq},a,\mathrm{MMM}}\)

Equivalent amplitude of MMM stress

\(R_\mathrm{M}\)

Interim invariant radius of Mohr

\(m_{a}\)

Macro-supporting effect factor

\(\sigma _1, \sigma _2 , \sigma _3\)

Principal stresses according to MMM

\(\varepsilon _a\)

Strain amplitude

\(\hat{{\sigma }}_{a,1}\)

Highest alternating principal stress

\(\eta \)

Dimensionless time

\(G_{\mathrm{MMM}}\)

Normalized stress gradient

\(\hat{{\varepsilon }}_\mathrm{H}\)

Linear elastic maximum (Hooke-) strain

\(\sigma _{\mathrm{loc}}^*\)

Local stress in equivalent tension specimen

Open image in new window

Lower Neuber stress level

\(\sigma _{\mathrm{eq},m}\)

Equivalent MMM mean stress

\(N_{\mathrm{alt}}\)

Iteration load cycle

\(\sigma _{-1,N}\)

Compression–tension fatigue limit for certain load cycle

\(\sigma _{\mathrm{eq},\mathrm{Mises}}\)

Equivalent von Mises stress

\(\sigma _x^o , \sigma _y^o , \sigma _z^o\)

Normal stresses from FEM, BEM calculation

\(\sigma \)

Stress

\(C_\mathrm{M}\)

Invariant center of Mohr

V

Sign function

\(n_{el}\)

\(\hbox {K}_{\mathrm{t}}\)\(\hbox {K}_{\mathrm{f}}\) ratio (micro-supporting factor)

\(k_a\)

Tension–shear endurance limit ratio

N

Load cycle

\(\hat{{\sigma }}_{o,1}\)

Highest maximum principal stress

\(\varphi \)

Constraint factor

\(\hat{{\sigma }}_\mathrm{H}\)

Linear elastic maximum (Hooke-) stress

\(\sigma _{\mathrm{loc}}\)

Local stress in notch

\(\varepsilon _{\mathrm{loc}}\)

Local strain in notch

Open image in new window

Lower Neuber strain level

\(\sigma _{\mathrm{eq},A,\mathrm{MMM}}\)

Sustainable equivalent amplitude of MMM stress

\(N_\mathrm{E}\)

Endurance load cycle

\(k_{aN}\)

Tension–shear limit ratio for certain load cycle

\(\sigma ^{\prime }\)

Stresses without supporting effects

\(\tau _{xy}^o, \tau _{xz}^o, \tau _{yz}^o\)

Shear stresses from FEM, BEM calculation

References

  1. 1.
    Federal Aviation Administration-Aviation Rulemaking Advisory Committee (FAA-ARAC).: Engine Windmilling Imbalance Loads. Final Report—Draft (1997)Google Scholar
  2. 2.
    European Aviation Safety Agency (EASA).: NPA for Engine Auxiliary Power Unit (APU) Failure Loads and Sustained Engine Windmilling (2007)Google Scholar
  3. 3.
    European Aviation Safety Agency.: Certification Specifications and Acceptable Means of Compliance for Large Aeroplanes CS-25 (2013)Google Scholar
  4. 4.
    Berger, C., Eulitz, K.-G., Heuler, P., Kotte, K.-L., Naundorf, H., Schuetz, W., Sonsino, C.M., Wimmer, A., Zenner, H.: Betriebsfestigkeit in Germany—an overview. Int. J. Fatigue 24, 603–625 (2002)CrossRefGoogle Scholar
  5. 5.
    You, B.-R., Lee, A.: A critical review on multiaxial fatigue assessment of metals. Int. J. Fatigue 18, 235–244 (1996)CrossRefGoogle Scholar
  6. 6.
    Papadopoulos, I.V., Davoli, P., Gorla, C., Filippini, M., Bernasconi, A.: A comparative study of multiaxial high-cycle fatigue criteria metals. Int. J. Fatigue 19, 219–235 (1997)CrossRefGoogle Scholar
  7. 7.
    Fatemi, A., Yang, L.: Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. Int. J. Fatigue 20, 9–34 (1998)CrossRefGoogle Scholar
  8. 8.
    Mughrabi, H.: Specific features and mechanisms of fatigue in the ultra-high-cycle regime. Int. J. Fatigue 28, 1501–1508 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Findlay, W.N.: Fatigue of metals under combination of stresses. In: Transaction of ASME, vol. 79. ASME (American Society of Mechanical Engineers), New York (1957)Google Scholar
  10. 10.
    Fatemi, A., Socie, D.F.: A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract. Eng. Mater. Struct. 11, 149–165 (1988)CrossRefGoogle Scholar
  11. 11.
    Papuga, J.: Mapping of fatigue damages—program shell of FE-calculation. Dissertation, CTU Prague (2005)Google Scholar
  12. 12.
    Van, K.D., Griveau, B., Message, O.: On a new multiaxial fatigue limit criterion: theory and application. Biaxial and Multiaxial Fatigue. EGF 3. Mechanical Engineering Publications. pp. 479–496 (1989)Google Scholar
  13. 13.
    Papadopoulos, I.V.: A new criterion for fatigue strength for out-of-phase bending and torsion of hard metals. Int. J. Fatigue 14, 377–384 (1994)CrossRefGoogle Scholar
  14. 14.
    Papadopoulos, I.V.: Long life fatigue under multiaxial loading. Int. J. Fatigue 23, 839–849 (2001)CrossRefGoogle Scholar
  15. 15.
    Ninic, D.: A stress-based multiaxial high-cycle fatigue damage criterion. Int. J. Fatigue 28, 108–113 (2006)CrossRefGoogle Scholar
  16. 16.
    Ninic, D., Stark, H.L.: A multiaxial fatigue damage function. Int. J. Fatigue 29, 533–548 (2007)CrossRefGoogle Scholar
  17. 17.
    Carpinteri, A., Spagnoli, A.: Multiaxial high-cycle fatigue criterion for hard metals. Int. J. Fatigue 23, 135–145 (2001)CrossRefGoogle Scholar
  18. 18.
    Dietmann, H.: Festigkeitsnachweis bei mehrachsiger- Schwingbeanspruchung. Konstruktion 25, 181–189 (1973). (in German)Google Scholar
  19. 19.
    Issler, L.: Festigkeitsverhalten metallischer Werkstoffe bei mehrachsiger phasenverschobener Schwingbeanspruchung. Techn.-Wiss. Ber. MPA Stuttgart. Heft 73-02. (in German) (1973)Google Scholar
  20. 20.
    Bhongbhibhat, T.: Festigkeitsverhalten von Stählen unter mehrachsiger phasenverschobener Schwingbeanspruchung mit unterschiedlichen Schwingformen und Frequenzen. Techn.-Wiss. Ber. MPA Stuttgart. Heft 86-01. (in German) (1986)Google Scholar
  21. 21.
    Novozhilov, V.V.: Theory of Elasticity. Pergamon, Oxford (1961)zbMATHGoogle Scholar
  22. 22.
    Issler, L.: Gültigkeitsgrenzen der Festigkeitshypothesen bei allgemeiner mehrachsiger Schwingbeanspruchung. Berichtsband 7. Sitzung DVM-Arbeitskreis Betriebsfestigkeit: pp. 295--314. (in German) (1981)Google Scholar
  23. 23.
    Haibach, E.: Betriebsfestigkeitslehre – Verfahren und Daten zur Bauteilberechnung. 3. Auflage. Springer, Heidelberg. (in German) (2006)Google Scholar
  24. 24.
    Mielke, S., Troost, A., El-Magd, E.: Strength of steels under biaxial synchronous and phase deviated oscillating normal stresses. Mater. Sci. Eng. Technol. 13, 1–7 (1982)Google Scholar
  25. 25.
    Troost, A., El-Magd, E.: Ermittlung der Versagensgrenzen zweiachsig schwingender Spannungszustände mit drei zeitabhängigen phasenverschobenen Spannungskoordinaten. DFG-Abschlussbericht. Tr 73/27-1. Deutsche Forschungsgemeinschaft. (in German) (1986)Google Scholar
  26. 26.
    Troost, A., Akin, O., Klubberg, F.: Dauerfestigkeitsverhalten metallischer Werkstoffe bei zweiachsiger Beanspruchung durch drei phasenverschobene schwingende Lastspannungen. Konstruktion 39 Heft 12, 479–488. (in German) (1987)Google Scholar
  27. 27.
    Troost, A., Akin, O., Klubberg, F.: Experimental data and calculated results about the fatigue endurance limit of metals under multiaxial alternating load. Mater. Sci. Eng. Technol. 23, 1–12 (1992). (in German)Google Scholar
  28. 28.
    Mertens, H., Pucknat, D.: Lebensdauer gekerbter metallischer Bauteile - Teil 1 und 2. Lebensdauer gekerbter metallischer Bauteile unter proportionaler und nichtproportionaler Beanspruchung. Konstruktion 1-2/2015. Springer-VDI, Düsseldorf. (in German) (2015)Google Scholar
  29. 29.
    Pucknat, D.: Berechnungs- und Bewertungsstrategie zur Schadenstoleranzanalyse komplexer Strukturen – Computation and evaluation strategy for damage tolerance analysis of complex structures. Dissertation, TU Berlin. (in German) (2015)Google Scholar
  30. 30.
    Analytical Strenght Assessment of Components---Made of Steel, Cast Iron and Aluminium Materials in Mechanical Engineering: FKM Guideline, 6th edn. VDMA, Frankfurt (2012)Google Scholar
  31. 31.
    Mertens, H., Hahn, M.: Vorhersage von Bauteilwöhlerlinien für Nennspannungskonzepte. Konstruktion 49, 31–37 (1997). (in German)Google Scholar
  32. 32.
    Kumar, V., German, M.D., Shih, C.F.: An engineering approach for elastic-plastic fracture analysis. Res. Proj. 1237–1 (NP 1931). Electric Power Res. Inst, Palo Alto, Cal (1981)Google Scholar
  33. 33.
    Rice, J.R.: A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. (ASME) 35, 379–386 (1968)CrossRefGoogle Scholar
  34. 34.
    Hoagland, R.G., Rosenfield, A.R., Gehlen, P.C., Hahn, G.T.: Fast fracture and crack arrest. Am. Soc. Test. Mater. ASTM STP 627, 177–202 (1977)Google Scholar
  35. 35.
    Newman, J.C.: A crack closure model for predicting fatigue crack growth under random loading. Am. Soc. Test. Mater. ASTM STP 748, 53–84 (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mechanical Engineering and Transport Systems, Institute of Engineering Design, Micro and Medical Technology, Institute of Aeronautics and Astronautics, Chair of Engineering Design and Product ReliabilityTechnical University of BerlinBerlinGermany

Personalised recommendations